How can I make some edges on my model sharper and others more rounded?
Mean Crease

I think the function you are looking for is called Mean Crease. It makes sharper edges when you are using a Subsurface modifier for example. Here's how it's used:  

  

If you want, once you have the edges selected in Edit mode, instead of manually adjusting the Mean Crease value as shown above, you can use the keyboard hotkeys for Mean Crease with the mouse:
SHIFTE -> drag the mouse